MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swopolem Structured version   Visualization version   GIF version

Theorem swopolem 5602
Description: Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypothesis
Ref Expression
swopolem.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
Assertion
Ref Expression
swopolem ((𝜑 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦,𝑌,𝑧   𝑧,𝑍
Allowed substitution hints:   𝑌(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem swopolem
StepHypRef Expression
1 swopolem.1 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
21ralrimivvva 3205 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
3 breq1 5146 . . . 4 (𝑥 = 𝑋 → (𝑥𝑅𝑦𝑋𝑅𝑦))
4 breq1 5146 . . . . 5 (𝑥 = 𝑋 → (𝑥𝑅𝑧𝑋𝑅𝑧))
54orbi1d 917 . . . 4 (𝑥 = 𝑋 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧𝑧𝑅𝑦)))
63, 5imbi12d 344 . . 3 (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧𝑧𝑅𝑦))))
7 breq2 5147 . . . 4 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
8 breq2 5147 . . . . 5 (𝑦 = 𝑌 → (𝑧𝑅𝑦𝑧𝑅𝑌))
98orbi2d 916 . . . 4 (𝑦 = 𝑌 → ((𝑋𝑅𝑧𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧𝑧𝑅𝑌)))
107, 9imbi12d 344 . . 3 (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧𝑧𝑅𝑌))))
11 breq2 5147 . . . . 5 (𝑧 = 𝑍 → (𝑋𝑅𝑧𝑋𝑅𝑍))
12 breq1 5146 . . . . 5 (𝑧 = 𝑍 → (𝑧𝑅𝑌𝑍𝑅𝑌))
1311, 12orbi12d 919 . . . 4 (𝑧 = 𝑍 → ((𝑋𝑅𝑧𝑧𝑅𝑌) ↔ (𝑋𝑅𝑍𝑍𝑅𝑌)))
1413imbi2d 340 . . 3 (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧𝑧𝑅𝑌)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌))))
156, 10, 14rspc3v 3638 . 2 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌))))
162, 15mpan9 506 1 ((𝜑 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144
This theorem is referenced by:  swoer  8776  swoord1  8777  swoord2  8778
  Copyright terms: Public domain W3C validator